On the Turing complexity of learning finite families of algebraic structures

TL;DR

This paper investigates the computational complexity of learning finite families of algebraic structures, showing that such learning can require non-computable resources and providing examples of learnable but non-computable learnable structures.

Contribution

It establishes the Turing degree requirements for learning finite families of structures and constructs examples illustrating the limits of computable learning.

Findings

Learning finite families can require non-computable oracles.

Any learnable finite family can be learned with an oracle for the Halting set.

There exist learnable families that no computable learner can learn.

Abstract

In previous work, we have combined computable structure theory and algorithmic learning theory to study which families of algebraic structures are learnable in the limit (up to isomorphism). In this paper, we measure the computational power that is needed to learn finite families of structures. In particular, we prove that, if a family of structures is both finite and learnable, then any oracle which computes the Halting set is able to achieve such a learning. On the other hand, we construct a pair of structures which is learnable but no computable learner can learn it.